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Quasi-isometric embeddings of symmetric spaces

David Fisher and Kevin Whyte

Geometry & Topology 22 (2018) 3049–3082

This paper opens the study of quasi-isometric embeddings of symmetric spaces. The main focus is on the case of equal and higher rank. In this context some expected rigidity survives, but some surprising examples also exist. In particular there exist quasi-isometric embeddings between spaces X and Y where there is no isometric embedding of X into Y . A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow–Morse lemma in higher rank. Typically this lemma is replaced by the quasiflat theorem, which says that the maximal quasiflat is within bounded distance of a finite union of flats. We improve this by showing that the quasiflat is in fact flat off of a subset of codimension 2.

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symmetric spaces, quasi-isometries, coarse geometry, rigidity
Mathematical Subject Classification 2010
Primary: 22E40, 53C24, 53C35
Received: 24 May 2017
Revised: 14 January 2018
Accepted: 4 March 2018
Published: 1 June 2018
Proposed: Bruce Kleiner
Seconded: Martin Bridson, Anna Wienhard
David Fisher
Department of Mathematics
Indiana University
Bloomington, IN
United States
Kevin Whyte
Department of Mathematics
University of Illinois at Chicago
Chicago, IL
United States